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We present the first results of three-dimensional (3D) numerical magnetohydrodynamic (MHD) simulations of the onset of magnetic reconnection via the tearing instability in dynamically thinning current sheets in the solar corona. In all our simulations, the onset of the non-linear tearing instability, which leads to the break-up of the thinning current sheet, does not occur until after the instability growth time becomes faster than the dynamic thinning time. Furthermore, as in previous 3D MHD simulations of static current sheets in the corona, for some parameters, the amount of magnetic shear is a fundamental switch-on parameter, which has consequences for coronal heating models. These results open up the possibility of using observable quantities of coronal current sheets to predict when they will break-up and release magnetic energy to power various energetic phenomena and/or heat the atmosphere.

We present the results of models of impulsively heated coronal loops using the 1-D hydrodynamic Adaptively Refined Godunov Solver (ARGOS) code. The impulsive heating events (which we refer to as "nanoflares") are modeled by discrete pulses of energy along the loop. We explore the occurrence of cold condensations due to the effective equivalent of thermal non-equilibrium (TNE) in loops with steady heating, and examine its dependence on nanoflare timing and intensity and also nanoflare location along the loop, including randomized distributions of nanoflares. We find that randomizing nanoflare distributions, both in time/intensity and location, diminishes the likelihood of condensations as compared to distributions with regularly occurring nanoflares with the same average properties. The usual criteria that condensations are favored for heating near loop footpoints and with high cadences are more strict for randomized (as opposed to regular) nanoflare distributions, and for randomized distributions the condensations stay in the loop for a shorter amount of time. That said, condensations can sometimes occur in cases where the average values of parameters (frequency or location) are beyond the critical limits above which condensations do not occur for corresponding steady, non-randomized values of those parameters. These properties of condensations occurring due to randomized heating can be used in the future to investigate diagnostics of coronal heating mechanisms.

We report results of an in-depth numerical investigation of three-dimensional projection effects which could influence the observed loop-like structures in an optically thin solar corona. Several archetypal emitting geometries are tested, including collections of luminous structures with circular cross-sections of fixed and random size, light-emitting structures with highly anisotropic cross-sections, as well as two-dimensional stochastic current density structures generated by fully-developed magnetohydrodynamic (MHD) turbulence. A comprehensive set of statistical signatures is used to compare the line of sight -integrated emission signals predicted by the constructed numerical models with the loop profiles observed by the extreme ultraviolet telescope onboard the flight 2.1 of the High-Resolution Coronal Imager (Hi-C). The results suggest that typical cross-sectional emission envelopes of the Hi-C loops are unlikely to have high eccentricity, and that the observed loops cannot be attributed to randomly oriented quasi-two dimensional emitting structures, some of which would produce anomalously strong optical signatures due to an accidental line-of-sight alignment expected in the coronal veil scenario \citep{malanushenko2022}. The possibility of apparent loop-like projections of very small (close to the resolution limit) or very large (comparable with the size of an active region) light-emitting sheets remains open, but the intermediate range of scales commonly associated with observed loop systems is most likely filled with true quasi-one dimensional (roughly axisymmetric) structures embedded into the three-dimensional coronal volume., Comment: 21 pages, 9 figures, 2 tables

The thickness of current sheets is extremely important, especially as it relates to the onset of fast magnetic reconnection. Onset determines how much magnetic free energy can build up in a field before it is explosively released. This has implications for many phenomena on the Sun and throughout the universe, including the heating of the solar corona. Significant effort has been devoted to the question of whether equilibrium current sheets in realistic geometries have finite or zero thickness. Using a simple force balance analysis, we show why current sheets without a guide field (2D) and with a guide field that is invariant in the guide field direction (2.5D) cannot be in equilibrium if they have both finite thickness and finite length. We then estimate the conditions under which the tension of a curved line-tied guide field can facilitate equilibrium in 3D sheets that are finite in all dimensions. Finally, we argue that some quasi-statically evolving current sheets undergoing slow stressing (e.g., when the coronal magnetic field is subjected to photospheric boundary driving) may reach a critical shear, at which point they lose equilibrium, spontaneously collapse, and reconnect. The critical shear is generally consistent with the heating requirements of solar active regions., Comment: 15 pages, 5 figures, accepted by Frontiers in Physics

FOXSI is a direct-imaging, hard X-ray (HXR) telescope optimized for solar flare observations. It detects hot plasma and energetic electrons in and near energy release sites in the solar corona via bremsstrahlung emission, measuring both spatial structure and particle energy distributions. It provides two orders of magnitude faster imaging spectroscopy than previously available, probing physically relevant timescales (<1s) never before accessible to address fundamental questions of energy release and efficient particle acceleration that have importance far beyond their solar application (e.g., planetary magnetospheres, flaring stars, accretion disks). FOXSI measures not only the bright chromospheric X-ray emission where electrons lose most of their energy, but also simultaneous emission from electrons as they are accelerated in the corona and propagate along magnetic field lines. FOXSI detects emission from high in the tenuous corona, where previous instruments have been blinded by nearby bright features and will fully characterizes the accelerated electrons and hottest plasmas as they evolve in energy, space, and time to solve the mystery of how impulsive energy release leads to solar eruptions, the primary drivers of space weather at Earth, and how those eruptions are energized and evolve., Comment: White paper submitted to the Decadal Survey for Solar and Space Physics (Heliophysics) 2024-2033; 14 pages, 4 figures, 1 table

It is essential that there be coordinated and co-optimized observations in X-rays, gamma-rays, and EUV during the peak of solar cycle 26 (~2036) to significantly advance our understanding of impulsive energy release in the corona. The open questions include: What are the physical origins of space-weather events? How are particles accelerated at the Sun? How is impulsively released energy transported throughout the solar atmosphere? How is the solar corona heated? Many of the processes involved in triggering, driving, and sustaining solar eruptive events -- including magnetic reconnection, particle acceleration, plasma heating, and energy transport in magnetized plasmas -- also play important roles in phenomena throughout the Universe. This set of observations can be achieved through a single flagship mission or, with foreplanning, through a combination of major missions (e.g., the previously proposed FIERCE mission concept)., Comment: White paper submitted to the Decadal Survey for Solar and Space Physics (Heliophysics) 2024-2033; 5 pages, 1 figure

Studying Doppler shifts provides deeper insights into the flow of mass and energy in the solar atmosphere. We perform a comprehensive measurement of Doppler shifts in the transition region and its center-to-limb variation (CLV) in the strong field regions ($|\textbf{B}| \geq$ 50 G) of 50 active regions (ARs), using the \ion{Si}{4} 1394~{\AA} line recorded by the Interface Region Imaging Spectrometer(IRIS). To locate the ARs and identify strong field regions, we have used the magnetograms obtained by the Helioseismic and Magnetic Imager (HMI). We find that in strong field regions, on average, all the ARs show mean redshifts ranging between 4{--}11~ km/s, which varies with ARs. These flows show a mild CLV, with sizable magnitudes at the limb and substantial scatter at the mid-longitude range. Our observations do not support the idea that redshifts in the lower transition region (T $<\sim$ 0.1 MK) are produced by field-aligned downflows as a result of impulsive heating and warrant alternative interpretation, such as downflow of type-\rm{II} spicules in the presence of a chromospheric wall created by cooler type-\rm{I} spicules., Comment: Accepted for publication in the Astrophysical Journal on 16th January 2023

Small-scale impulsive events, known as nanoflares, are thought to be one of the prime candidates that can keep the solar corona hot at its multi-million Kelvin temperature. Individual nanoflares are difficult to detect with the current generation instruments; however, their presence can be inferred through indirect techniques such as a Differential Emission Measure (DEM) analysis. Here we employ this technique to investigate the possibility of nanoflare heating of the quiet corona during the minimum of solar cycle 24. During this minimum, active regions (ARs) were absent on the solar-disk for extended periods. In the absence of ARs, X-ray bright points (XBP) are the dominant contributor to disk-integrated X-rays. We estimate the DEM of the XBPs using observations from the Solar X-ray Monitor (XSM) onboard the Chandrayaan-2 orbiter and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamic Observatory. XBPs consist of small-scale loops associated with bipolar magnetic fields. We simulate such XBP loops using the EBTEL hydrodynamic code. The lengths and magnetic field strengths of these loops are obtained through a potential field extrapolation of the photospheric magnetogram. Each loop is assumed to be heated by random nanoflares having an energy that depends on the loop properties. The composite nanoflare energy distribution for all the loops has a power-law slope close to -2.5. The simulation output is then used to obtain the integrated DEM. It agrees remarkably well with the observed DEM at temperatures above 1 MK, suggesting that the nanoflare distribution, as predicted by our model, can explain the XBP heating., Comment: Submitted in the Astrophysical Journal, 20 pages, 10 figures

To facilitate the study of solar active regions and flaring loops, we have created a modeling framework, the freely distributed GX Simulator IDL package, that combines 3D magnetic and plasma structures with thermal and non-thermal models of the chromosphere, transition region, and corona. The package has integrated tools to visualize the model data cubes, compute multi-wavelength emission maps from them, and quantitatively compare the resulting maps with observations. Its object-based modular architecture, which runs on Windows, Mac, and Unix/Linux platforms, offers capabilities that include the ability to either import 3D density and temperature distribution models, or to assign numerically defined coronal or chromospheric temperatures and densities, or their distributions to each individual voxel. The application integrates FORTRAN and C++ libraries for fast calculation of radio emission (free-free, gyroresonance, and gyrosynchrotron emission) along with soft and hard X-ray and EUV codes developed in IDL. To facilitate the creation of models, we have developed a fully automatic model production pipeline that downloads the required SDO/HMI vector magnetic field data and (optionally) the contextual SDO/AIA images, performs potential or nonlinear force free field extrapolations, populates the magnetic field skeleton with parameterized heated plasma coronal models that assume either steady-state or impulsive plasma heating, and generates non-LTE density and temperature distribution models of the chromosphere that are constrained by photospheric measurements. The standardized models produced by this pipeline may be further customized through a set of interactive tools provided by the graphical user interface. Here we describe the GX Simulator framework and its applications.

It is extremely difficult to simulate the details of coronal heating and also make meaningful predictions of the emitted radiation. Thus, testing realistic models with observations is a major challenge. Observational signatures of coronal heating depend crucially on radiation, thermal conduction, and the exchange of mass and energy with the transition region and chromosphere below. Many magnetohydrodynamic simulation studies do not include these effects, opting instead to devote computational resources to the magnetic aspects of the problem. We have developed a simple method of accounting approximately for the missing effects. It is applied to the simulation output post facto and therefore may be a valuable tool for many studies. We have used it to predict the emission from a model corona that is driven by vortical boundary motions meant to represent photospheric convection. We find that individual magnetic strands experience short-term brightenings, both scattered throughout the computational volume and in localized clusters. The former may explain the diffuse component of the observed corona, while the latter may explain bright coronal loops. Several observed properties of loops are reproduced reasonably well: width, lifetime, and quasi-circular cross-section (aspect ratio not large). Our results lend support to the idea that loops are multi-stranded structures heated by "storms" of nanoflares., Comment: Accepted by the Astrophysical Journal; 25 pages, 15 figures, 2 movies

The nanoflare model of coronal heating is one of the most successful scenarios to explain, within a single framework, the diverse set of coronal observations available with the present instrument resolutions. The model is based on the idea that the coronal structure is formed by elementary magnetic strands which are tangled and twisted by the displacement of their photospheric footpoints by convective motions. These displacements inject magnetic stress between neighbor strands that promotes current sheet formation, reconnection, plasma heating, and possibly also particle acceleration. Among other features, the model predicts the ubiquitous presence of plasma flows at different temperatures. These flows should, in principle, produce measurable effects on observed spectral lines in the form of Doppler-shifts, line asymmetries and non-thermal broadenings. In this work we use a Two-Dimensional Cellular Automaton Model (2DCAM) developed in previous works, in combination with the Enthalpy Based Thermal Evolution of Loops (EBTEL) model, to analyze the effect of nanoflare heating on a set of known EUV spectral lines. We find that the complex combination of the emission from plasmas at different temperatures, densities and velocities, in simultaneously evolving unresolved strands, produces characteristic properties in the constructed synthetic lines, such as Doppler-shifts and non-thermal velocities up to tens of km s$^{-1}$ for the higher analyzed temperatures. Our results might prove useful to guide future modeling and observations, in particular, regarding the new generation of proposed instruments designed to diagnose plasmas in the 5 to 10 MK temperature range., Comment: Accepted for publication in The Astrophysical Journal

Heliophysics theory and modeling build understanding from fundamental principles to motivate, interpret, and predict observations. Together with observational analysis, they constitute a comprehensive scientific program in heliophysics. As observations and data analysis become increasingly detailed, it is critical that theory and modeling develop more quantitative predictions and iterate with observations. Advanced theory and modeling can inspire and greatly improve the design of new instruments and increase their chance of success. In addition, in order to build physics-based space weather forecast models, it is important to keep developing and testing new theories, and maintaining constant communications with theory and modeling. Maintaining a sustainable effort in theory and modeling is critically important to heliophysics. We recommend that all funding agencies join forces and consider expanding current and creating new theory and modeling programs--especially, 1. NASA should restore the HTMS program to its original support level to meet the critical needs of heliophysics science; 2. a Strategic Research Model program needs to be created to support model development for next-generation basic research codes; 3. new programs must be created for addressing mission-critical theory and modeling needs; and 4. enhanced programs are urgently required for training the next generation of theorists and modelers., Comment: White paper submitted to Heliophysics 2024 Decadal Survey

Recent high-resolution imaging and spectroscopic observations have generated renewed interest in spicules' role in explaining the hot corona. Some studies suggest that some spicules, often classified as type II, may provide significant mass and energy to the corona. Here we use numerical simulations to investigate whether such spicules can produce the observed coronal emission without any additional coronal heating agent. Model spicules consisting of a cold body and hot tip are injected into the base of a warm ($0.5$ MK) equilibrium loop with different tip temperatures and injection velocities. Both piston- and pressure-driven shocks are produced. We find that the hot tip cools rapidly and disappears from coronal emission lines such as Fe XII $195$ and Fe XIV $274$. Prolonged hot emission is produced by pre-existing loop material heated by the shock and by thermal conduction from the shock. However, the shapes and Doppler shifts of synthetic line profiles show significant discrepancies with observations. Furthermore, spatially and temporally averaged intensities are extremely low, suggesting that if the observed intensities from the quiet Sun and active regions were solely due to type II spicules, one to several orders of magnitude more spicules would be required than have been reported in the literature. This conclusion applies strictly to the ejected spicular material. We make no claims about emissions connected with waves or coronal currents that may be generated during the ejection process and heat the surrounding area., Comment: Accepted for publication in the Astrophysical Journal, 16 pages, 17 figures

Results from 2.5D and 3D studies of the onset and development of the tearing instability are presented, using high fidelity resistive MHD simulations. A limited parameter study of the strength of the reconnecting field (or shear angle) was performed. An initially simple 1D equilibrium was used, consisting of a modified force-free current sheet, with periodic boundary conditions in all directions. In all cases, the linear and non-linear evolution led to a primary current sheet between two large flux ropes. The global reconnection rate during this later stage was analyzed in all simulations. It was found that in 2.5D the primary current sheet fragmented due to plasmoids, and as expected, the global reconnection rate, calculated using multiple methods, increases with the strength of the reconnecting field due to a stronger Alfv\'{e}n speed. In 3D, the presence of interacting oblique modes of the tearing instability complicates the simple 2.5D picture, entangling the magnetic field of the inflow and introducing a negative effect on the reconnection rate. The two competing effects of stronger Alfv\'{e}n speed and entangling, which both increase with the strength of the reconnecting field, resulted in a decrease in the reconnection rate with increasing reconnecting field. For all simulations, the 3D rates were less than in 2.5D, but suggest that as one goes to weak reconnecting field (or strong guide field), the system becomes more 2.5D like and the 2.5D and 3D rates converge. These results have relevance to situations like nano-flare heating and flare current sheets in the corona., Comment: 15 pages, 10 figures

Plasma-filled loop structures are common in the solar corona. Because detailed modeling of the dynamical evolution of these structures is computationally costly, an efficient method for computing approximate but quick physics-based solutions is to rely on space integrated 0D simulations. The enthalpy-based thermal evolution of loops EBTEL framework is a commonly used method to study the exchange of mass and energy between the corona and transition region. EBTEL solves for density, temperature, and pressure averaged over the coronal part of the loop, the velocity at the coronal base, and the instantaneous differential emission measure distribution in the transition region. The current single-fluid version of the code, EBTEL2, assumes that at all stages the flows are subsonic. However, sometimes the solutions show the presence of supersonic flows during the impulsive phase of heat input. It is thus necessary to account for this effect. Here, we upgrade EBTEL2 to EBTEL3 by including the kinetic energy term in the Navier-Stokes equation. We compare the solutions from EBTEL3 with those obtained using EBTEL2 as well as the state-of-the-art field-aligned hydrodynamics code HYDRAD. We find that the match in pressure between EBTEL3 and HYDRAD is better than that between EBTEL2 and HYDRAD. Additionally, the velocities predicted by EBTEL3 are in close agreement with those obtained with HYDRAD when the flows are subsonic. However, EBTEL3 solutions deviate substantially from HYDRAD's when the latter predicts supersonic flows. Using the mismatches in the solution, we propose a criterion to determine the conditions under which EBTEL can be used to study flows in the system., Comment: Accepted for publication in ApJ

There is a wide consensus that the ubiquitous presence of magnetic reconnection events and the associated impulsive heating (nanoflares) is a strong candidate for solving the solar coronal heating problem. Whether nanoflares accelerate particles to high energies like full-sized flares is unknown. We investigate this question by studying the type III radio bursts that the nanoflares may produce on closed loops. The characteristic frequency-drifts that type III bursts exhibit can be detected using a novel application of the time-lag technique developed by Viall & Klimchuk (2012) even when there are multiple overlapping bursts. We present a simple numerical model that simulates the expected radio emission from nanoflares in an active region (AR), which we use to test and calibrate the technique. We find that in the case of closed loops the frequency spectrum of type III bursts is expected to be extremely steep such that significant emission is produced at a given frequency only for a rather narrow range of loop lengths. We also find that the signature of bursts in the time-lag signal diminishes as: (1)the variety of participating loops within that range increases; (2)the occurrence rate of bursts increases; (3) the duration of bursts increases; and (4) the brightness of the bursts decreases relative to noise. In addition, our model suggests a possible origin of type I bursts as a natural consequence of type III emission in a closed-loop geometry.

We argue that the magnetically closed corona evolves primarily quasi-statically, punctuated by many localized bursts of activity associated with magnetic reconnection at a myriad of small current sheets. The sheets form by various processes that do not involve a traditional turbulent cascade whereby energy flows losslessly through a continuum of spatial scales starting from the large scale of the photospheric driving. If such an inertial range is a defining characteristic of turbulence, then the magnetically closed corona is not a turbulent system. It nonetheless has a complex structure that bears no direct relationship to the pattern of driving., Comment: accepted by Frontiers in Astronomy and Space Sciences - Space Physics

We derive the non-thermal velocities (NTVs) in the transition region of an active region using the \ion{Si}{4}~1393.78~{\AA} line observed by the Interface Region Imaging Spectrograph (IRIS) and compare them with the line-of-sight photospheric magnetic fields obtained by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). The active region consists of two strong field regions with opposite polarity, separated by a weak field corridor, that widened as the active region evolved. The means of the NTV distributions in strong-field regions (weak field corridors) range between $\sim$18{--}20 (16{--}18)~km~s$^{-1}$, albeit the NTV maps show much larger range. In addition, we identify a narrow lane in the middle of the corridor with significantly reduced NTV. The NTVs do not show a strong center-to-limb variation, albeit somewhat larger values near the disk center. The NTVs are well correlated with redshifts as well as line intensities. The results obtained here and those presented in our companion paper on Doppler shifts suggest two populations of plasma in the active region emitting in \ion{Si}{4}. The first population exists in the strong field regions and extends partway into the weak field corridor between them. We attribute this plasma to spicules heated to $\sim$0.1 MK (often called type II spicules). They have a range of inclinations relative to vertical. The second population exists in the center of the corridor, is relatively faint, and has smaller velocities, likely horizontal. These results provide further insights into the heating of the transition region., Comment: 23 pages, 8 figures, accepted for publication in The Astrophysical Journal

We investigate the relative contributions from the transition region and corona of coronal loops observed by the Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO). Using EBTEL (Enthalpy-Based Thermal Evolution of Loops) hydrodynamic simulations, we model loops with multiple lengths and energy fluxes heated randomly by events drawn from power-law distributions with different slopes and minimum delays between events to investigate how each of these parameters influences observable loop properties. We generate AIA intensities from the corona and transition region for each realization. The variations within and between models generated with these different parameters illustrate the sensitivity of narrowband imaging to the details of coronal heating. We then analyze the transition region and coronal emission from a number of observed active regions and find broad agreement with the trends in the models. In both models and observations, the transition region brightness is significant, often greater than the coronal brightness in all six "coronal" AIA channels. We also identify an inverse relationship, consistent with heating theories, between the slope of the differential emission measure (DEM) coolward of the peak temperature and the observed ratio of coronal to transition region intensity. These results highlight the use of narrowband observations and the importance of properly considering the transition region in investigations of coronal heating., Comment: Accepted for publication in ApJ. 16 pages, 10 figures, 4 tables

Coronal loops reveal crucial information about the nature of both coronal magnetic fields and coronal heating. The shape of the corresponding flux tube cross section and how it varies with position are especially important properties. They are a direct indication of the expansion of the field and of the cross-field spatial distribution of the heating. We have studied 20 loops using high spatial resolution observations from the first flight of the Hi-C rocket experiment, measuring the intensity and width as a function of position along the loop axis. We find that intensity and width tend to either be uncorrelated or to have a direct dependence, such that they increase or decrease together. This implies that the flux tube cross sections are approximately circular under the assumptions that the tubes have non-negligible twist and that the plasma emissivity is approximately uniform along the magnetic field. The shape need not be a perfect circle and the emissivity need not be uniform within the cross section, but sub-resolution patches of emission must be distributed quasi-uniformly within an envelope that has an aspect ratio of order unity. This raises questions about the suggestion that flux tubes expand with height, but primarily in the line-of-sight direction so that the corresponding (relatively noticeable) loops appear to have roughly uniform width, a long-standing puzzle. It also casts doubt on the idea that most loops correspond to simple warped sheets, although we leave open the possibility of more complex manifold structures., Comment: 27 pages, 9 figures, accepted by Astrophysical Journal

Magnetic reconnection, a fundamentally important process in many aspects of astrophysics, is believed to be initiated by the tearing instability of an electric current sheet, a region where magnetic field abruptly changes direction and electric currents build up. Recent studies have suggested that the amount of magnetic shear in these structures is a critical parameter for the switch-on nature of magnetic reconnection in the solar atmosphere, at fluid spatial scales much larger than kinetic scales. We present results of simulations of reconnection in 3D current sheets with conditions appropriate to the solar corona. Using high-fidelity simulations, we follow the evolution of the linear and non-linear 3D tearing instability, leading to reconnection. We find that, depending on the parameter space, magnetic shear can play a vital role in the onset of significant energy release and heating via non-linear tearing. Two regimes in our study exist, dependent on whether the current sheet is longer or shorter than the wavelength of the fastest growing parallel mode (in the corresponding infinite system), thus determining whether sub-harmonics are present in the actual system. In one regime, where the fastest growing parallel mode has sub-harmonics, the non-linear interaction of these sub-harmonics and the coalescence of 3D plasmoids dominates the non-linear evolution, with magnetic shear playing only a weak role in the amount of energy released. In the second regime, where the fastest growing parallel mode has no-sub-harmonics, then only strongly sheared current sheets, where oblique mode are strong enough to compete with the dominant parallel mode, show any significant energy release. We expect both regimes to exist on the Sun, and so our results have important consequences for the the question of reconnection onset in different solar physics applications., Comment: 11 pages9 figures

For some forms of steady heating, coronal loops are in a state of thermal nonequilibrium and evolve in a manner that includes accelerated cooling, often resulting in the formation of a cold condensation. This is frequently confused with thermal instability, but the two are in fact fundamentally different. We explain the distinction and discuss situations where they may be interconnected. Large-amplitude perturbations, perhaps associated with MHD waves, likely play a role in explaining phenomena that have been attributed to thermal nonequilibrium but also seem to require cross-field communication., Comment: accepted by Solar Physics 16 pages, 1 figure

A comprehensive understanding of the structure of Doppler motions in transition region including the center-to-limb variation and its relationship with the magnetic field structure is vital for the understanding of mass and energy transfer in the solar atmosphere. In this paper, we have performed such a study in an active region using the Si IV 1394~{\AA} emission line recorded by the Interface Region Imaging Spectrograph (IRIS) and the line-of-sight photospheric magnetic field obtained by the Helioseismic and Magnetic Imager (HMI) on-board the Solar Dynamics Observatory (SDO). The active region has two opposite polarity strong field regions separated by a weak field corridor, which widened as the active region evolved. On average the strong field regions (corridor) show(s) redshifts of 5{--}10 (3{--}9)~km~s$^{-1}$ (depending on the date of observation). There is, however, a narrow lane in the middle of the corridor with near-zero Doppler shifts at all disk positions, suggesting that any flows there are very slow. The Doppler velocity distributions in the corridor seem to have two components---a low velocity component centered near 0 km/s and a high velocity component centered near 10~km~s$^{-1}$. The high velocity component is similar to the velocity distributions in the strong field regions, which have just one component. Both exhibit a small center-to limb variation and seem to come from the same population of flows. To explain these results, we suggest that the emission from the lower transition region comes primarily from warm type II spicules, and we introduce the idea of a `chromospheric wall'---associated with classical cold spicules---to account for a diminished center-to-limb variation., Comment: 15 pages, 16 figures

One of the greatest challenges in solar physics is understanding the heating of the Sun's corona. Most theories for coronal heating postulate that free energy in the form of magnetic twist/stress is injected by the photosphere into the corona where the free energy is converted into heat either through reconnection or wave dissipation. The magnetic helicity associated with the twist/stress, however, is expected to be conserved and appear in the corona. In previous work we showed that helicity associated with the small-scale twists undergoes an inverse cascade via stochastic reconnection in the corona, and ends up as the observed large-scale shear of filament channels. Our ``helicity condensation'' model accounts for both the formation of filament channels and the observed smooth, laminar structure of coronal loops. In this paper, we demonstrate, using helicity- and energy-conserving numerical simulations of a coronal system driven by photospheric motions, that the model also provides a natural mechanism for heating the corona. We show that the heat generated by the reconnection responsible for the helicity condensation process is sufficient to account for the observed coronal heating. We study the role that helicity injection plays in determining coronal heating and find that, crucially, the heating rate is only weakly dependent on the net helicity preference of the photospheric driving. Our calculations demonstrate that motions with 100\% helicity preference are least efficient at heating the corona; those with 0\% preference are most efficient. We discuss the physical origins of this result and its implications for the observed corona., Comment: Accepted by ApJ

Thermal non-equilibrium (TNE) is a fascinating situation that occurs in coronal magnetic flux tubes (loops) for which no solution to the steady-state fluid equations exists. The plasma is constantly evolving even though the heating that produces the hot temperatures does not. This is a promising explanation for isolated phenomena such as prominences, coronal rain, and long-period pulsating loops, but it may also have much broader relevance. As known for some time, TNE requires that the heating be both (quasi) steady and concentrated at low coronal altitudes. Recent studies indicate that asymmetries are also important, with large enough asymmetries in the heating and/or cross-sectional area resulting in steady flow rather than TNE. Using reasonable approximations, we have derived two formulae for quantifying the conditions necessary for TNE. As a rough rule of thumb, the ratio of apex to footpoint heating rates must be less than about 0.1, and asymmetries must be less than about a factor of 3. The precise values are case dependent. We have tested our formulae with 1D hydrodynamic loop simulations and find a very acceptable agreement. These results are important for developing physical insight about TNE and assessing how widespread it may be on the Sun., Comment: 38 pages, 9 figures, 2 tables, accepted by Astrophysical Journal

Much evidence suggests that the solar corona is heated impulsively, meaning that nanoflares may be ubiquitous in quiet and active regions (ARs). Hard X-ray (HXR) observations with unprecedented sensitivity $>$3~keV are now enabled by focusing instruments. We analyzed data from the \textit{Focusing Optics X-ray Solar Imager (FOXSI)} rocket and the \textit{Nuclear Spectroscopic Telescope Array (NuSTAR)} spacecraft to constrain properties of AR nanoflares simulated by the EBTEL field-line-averaged hydrodynamics code. We generated model X-ray spectra by computing differential emission measures for homogeneous nanoflare sequences with heating amplitudes $H_0$, durations $\tau$, delay times between events $t_N$, and filling factors $f$. The single quiescent AR observed by \textit{FOXSI-2} on 2014 December 11 is well fit by nanoflare sequences with heating amplitudes 0.02 erg cm$^{-3}$ s$^{-1}$ $<$ $H_0$ $<$ 13 erg cm$^{-3}$ s$^{-1}$ and a wide range of delay times and durations. We exclude delays between events shorter than $\sim$900 s at the 90\% confidence level for this region. Three of five regions observed by {\nustar} on 2014 November 1 are well fit by homogeneous nanoflare models, while two regions with higher fluxes are not. Generally, the {\nustar} count spectra are well fit by nanoflare sequences with smaller heating amplitudes, shorter delays, and shorter durations than the allowed \textit{FOXSI-2} models. These apparent discrepancies are likely due to differences in spectral coverage between the two instruments and intrinsic differences among the regions. Steady heating ($t_N$ = $\tau$) was ruled out with $>$99\% confidence for all regions observed by either instrument., Comment: 17 pages, 21 figures. Accepted for publication in The Astrophysical Journal

We report on high-resolution imaging and spectral observations of eruptions of a spiral structure in the transition region, which were taken with the Interface Region Imaging Spectrometer (IRIS), the Atmospheric Imaging Assembly (AIA) and the Helioseismic and Magnetic Imager (HMI). The eruption coincided with the appearance of two series of jets, with velocities comparable to the Alfv\'en speeds in their footpoints. Several pieces of evidence of magnetic braiding in the eruption are revealed, including localized bright knots, multiple well-separated jet threads, transition region explosive events and the fact that all these three are falling into the same locations within the eruptive structures. Through analysis of the extrapolated three-dimensional magnetic field in the region, we found that the eruptive spiral structure corresponded well to locations of twisted magnetic flux tubes with varying curl values along their lengths. The eruption occurred where strong parallel currents, high squashing factors, and large twist numbers were obtained. The electron number density of the eruptive structure is found to be $\sim3\times10^{12}$ cm$^{-3}$, indicating that significant amount of mass could be pumped into the corona by the jets. Following the eruption, the extrapolations revealed a set of seemingly relaxed loops, which were visible in the AIA 94 \AA\ channel indicating temperatures of around 6.3 MK. With these observations, we suggest that magnetic braiding could be part of the mechanisms explaining the formation of solar eruption and the mass and energy supplement to the corona., Comment: 11 figs, accepted for publication in ApJ

Numerous observations have revealed that power-law distributions are ubiquitous in energetic solar processes. Hard X-rays, soft X-rays, extreme ultraviolet radiation, and radio waves all display power-law frequency distributions. Since magnetic reconnection is the driving mechanism for many energetic solar phenomena, it is likely that reconnection events themselves display such power-law distributions. In this work, we perform numerical simulations of the solar corona driven by simple convective motions at the photospheric level. Using temperature changes, current distributions, and Poynting fluxes as proxies for heating, we demonstrate that energetic events occurring in our simulation display power-law frequency distributions, with slopes in good agreement with observations. We suggest that the braiding-associated reconnection in the corona can be understood in terms of a self-organized criticality model driven by convective rotational motions similar to those observed at the photosphere., Comment: Accepted by ApJ

This is a review of the observational and theoretical evidence for nanoflare heating of the magnetically-closed corona., Comment: This is a section for a mega-paper entitled "Achievements of Hinode in the First Ten Years." The invited speakers of the Hinode 10 Conference in Nagoya were asked to each write a section. The contribution should be cited as Klimchuk, J. A. and Hinode Review Team, "Achievements of Hinode in the First Ten Years," 2017, PASJ, submitted

Even in the absence of resolved flares, the corona is heated to several million degrees. However, despite its importance for the structure, dynamics, and evolution of the solar atmosphere, the origin of this heating remains poorly understood. Several observational and theoretical considerations suggest that the heating is driven by small, impulsive energy bursts which could be Parker-style "nanoflares" (Parker 1988) that arise via reconnection within the tangled and twisted coronal magnetic field. The classical "smoking gun" (Klimchuk 2009; Cargill et al. 2013) for impulsive heating is the direct detection of widespread hot plasma (T > 6 MK) with a low emission measure. In recent years there has been great progress in the development of Transition Edge Sensor (TES) X-ray microcalorimeters that make them more ideal for studying the Sun. When combined with grazing-incidence focusing optics, they provide direct spectroscopic imaging over a broad energy band (0.5 to 10 keV) combined with extremely impressive energy resolution in small pixels, as low as 0.7 eV (FWHM) at 1.5 keV (Lee 2015), and 1.56 eV (FWHM) at 6 keV (Smith 2012), two orders of magnitude better than the current best traditional solid state photon-counting spectrometers. Decisive observations of the hot plasma associated with nanoflare models of coronal heating can be provided by new solar microcalorimeters. These measurements will cover the most important part of the coronal spectrum for searching for the nanoflare-related hot plasma and will characterize how much nanoflares can heat the corona both in active regions and the quiet Sun. Finally, microcalorimeters will enable to study all of this as a function of time and space in each pixel simultaneously a capability never before available., Comment: Next Generation Solar Physics Mission, 2 figures

How impulsive magnetic energy release leads to solar eruptions and how those eruptions are energized and evolve are vital unsolved problems in Heliophysics. The standard model for solar eruptions summarizes our current understanding of these events. Magnetic energy in the corona is released through drastic restructuring of the magnetic field via reconnection. Electrons and ions are then accelerated by poorly understood processes. Theories include contracting loops, merging magnetic islands, stochastic acceleration, and turbulence at shocks, among others. Although this basic model is well established, the fundamental physics is poorly understood. HXR observations using grazing-incidence focusing optics can now probe all of the key regions of the standard model. These include two above-the-looptop (ALT) sources which bookend the reconnection region and are likely the sites of particle acceleration and direct heating. The science achievable by a direct HXR imaging instrument can be summarized by the following science questions and objectives which are some of the most outstanding issues in solar physics (1) How are particles accelerated at the Sun? (1a) Where are electrons accelerated and on what time scales? (1b) What fraction of electrons is accelerated out of the ambient medium? (2) How does magnetic energy release on the Sun lead to flares and eruptions? A Focusing Optics X-ray Solar Imager (FOXSI) instrument, which can be built now using proven technology and at modest cost, would enable revolutionary advancements in our understanding of impulsive magnetic energy release and particle acceleration, a process which is known to occur at the Sun but also throughout the Universe., Comment: Next Generation Solar Physics Mission white paper, 2 figures

We use the cellular automaton model described in L\'opez Fuentes \& Klimchuk (2015, ApJ, 799, 128) to study the evolution of coronal loop plasmas. The model, based on the idea of a critical misalignment angle in tangled magnetic fields, produces nanoflares of varying frequency with respect to the plasma cooling time. We compare the results of the model with active region (AR) observations obtained with the Hinode/XRT and SDO/AIA instruments. The comparison is based on the statistical properties of synthetic and observed loop lightcurves. Our results show that the model reproduces the main observational characteristics of the evolution of the plasma in AR coronal loops. The typical intensity fluctuations have an amplitude of 10 to 15\% both for the model and the observations. The sign of the skewness of the intensity distributions indicates the presence of cooling plasma in the loops. We also study the emission measure (EM) distribution predicted by the model and obtain slopes in log(EM) versus log(T) between 2.7 and 4.3, in agreement with published observational values., Comment: Paper 2 of 2: Model comparison with observations. Accepted for publication in ApJ

We study a 2D cellular automaton (CA) model for the evolution of coronal loop plasmas. The model is based on the idea that coronal loops are made of elementary magnetic strands that are tangled and stressed by the displacement of their footpoints by photospheric motions. The magnetic stress accumulated between neighbor strands is released in sudden reconnection events or nanoflares that heat the plasma. We combine the CA model with the Enthalpy Based Thermal Evolution of Loops (EBTEL) model to compute the response of the plasma to the heating events. Using the known response of the XRT telescope on board Hinode we also obtain synthetic data. The model obeys easy to understand scaling laws relating the output (nanoflare energy, temperature, density, intensity) to the input parameters (field strength, strand length, critical misalignment angle). The nanoflares have a power-law distribution with a universal slope of -2.5, independent of the input parameters. The repetition frequency of nanoflares, expressed in terms of the plasma cooling time, increases with strand length. We discuss the implications of our results for the problem of heating and evolution of active region coronal plasmas., Comment: Paper 1 of 2: Model description. \bibitem[L{\'o}pez Fuentes \& Klimchuk(2015)]{2015ApJ...799..128L} L{\'o}pez Fuentes, M., \& Klimchuk, J.~A.\ 2015, \apj, 799, 128

Among the many ways of investigating coronal heating, the time lag method of Viall & Klimchuk (2012) is becoming increasingly prevalent as an analysis technique complementary to those traditionally used. The time lag method cross correlates light curves at a given spatial location obtained in spectral bands that sample different temperature plasmas. It has been used most extensively with data from the Atmospheric Imaging Assembly on the Solar Dynamics Observatory. We have previously applied the time lag method to entire active regions and surrounding quiet Sun and create maps of the results (Viall & Klimchuk 2012; Viall & Klimchuk 2015). We find that the majority of time lags are consistent with the cooling of coronal plasma that has been impulsively heated. Additionally, a significant fraction of the map area has a time lag of zero. This does not indicate a lack of variability. Rather, strong variability must be present, and it must occur in phase in the different channels. We have shown previously that these zero time lags are consistent with the transition region response to coronal nanoflares (Viall & Klimchuk 2015; Bradshaw & Viall 2016), but other explanations are possible. A common misconception is that the zero time lag indicates steady emission resulting from steady heating. Using simulated and observed light curves, we demonstrate here that highly correlated light curves at zero time lag are not compatible with equilibrium solutions. Such light curves can only be created by evolution., Comment: 10 pages, 3 figures. Accepted to ApJ July 5 2016

The million degree plasma of the solar corona must be supplied by the underlying layers of the atmosphere. The mechanism and location of energy release, and the precise source of coronal plasma, remain unresolved. In earlier work we pursued the idea that warm plasma is supplied to the corona via direct heating of the chromosphere by nanoflares, contrary to the prevailing belief that the corona is heated in-situ and the chromosphere is subsequently energized and ablated by thermal conduction. We found that single (low-frequency) chromospheric nanoflares could not explain the observed intensities, Doppler-shifts, and red/blue asymmetries in Fe XII and XIV emission lines. In the present work we follow up on another suggestion that the corona could be powered by chromospheric nanoflares that repeat on a timescale substantially shorter than the cooling/draining timescale. That is, a single magnetic strand is re-supplied with coronal plasma before the existing plasma has time to cool and drain. We perform a series of hydrodynamic experiments and predict the Fe XII and XIV line intensities, Doppler-shifts, and red/blue asymmetries. We find that our predicted quantities disagree dramatically with observations and fully developed loop structures cannot be created by intermediate- or high-frequency chromospheric nanoflares. We conclude that the mechanism ultimately responsible for producing coronal plasma operates above the chromosphere, but this does not preclude the possibility of a similar mechanism powering the chromosphere; extreme examples of which may be responsible for heating chromospheric plasma to transition region temperatures (e.g. type II spicules).

The detailed shapes of spectral line profiles provide valuable information about the emitting plasma, especially when the plasma contains an unresolved mixture of velocities, temperatures, and densities. As a result of finite spectral resolution, the intensity measured by a spectrometer is the average intensity across a wavelength bin of non-zero size. It is assigned to the wavelength position at the center of the bin. However, the actual intensity at that discrete position will be different if the profile is curved, as it invariably is. Standard fitting routines (spline, Gaussian, etc.) do not account for this difference, and this can result in significant errors when making sensitive measurements. Detection of asymmetries in solar coronal emission lines is one example. Removal of line blends is another. We have developed an iterative procedure that corrects for this effect. It can be used with any fitting function, but we employ a cubic spline in a new analysis routine called Intensity Conserving Spline Interpolation (ICSI). As the name implies, it conserves the observed intensity within each wavelength bin, which ordinary fits do not. Given the rapid convergence, speed of computation, and ease of use, we suggest that ICSI be made a standard component of the processing pipeline for spectroscopic data., Comment: 9 pages, 7 figures, 1 table; submitted to Astrophysical Journal (revised version)

We highlight ten key aspects of coronal heating that must be understood before we can consider the problem to be solved. (1) All coronal heating is impulsive. (2) The details of coronal heating matter. (3) The corona is filled with elemental magnetic stands. (4) The corona is densely populated with current sheets. (5) The strands must reconnect to prevent an infinite buildup of stress. (6) Nanoflares repeat with different frequencies. (7) What is the characteristic magnitude of energy release? (8) What causes the collective behavior responsible for loops? (9) What are the onset conditions for energy release? (10) Chromospheric nanoflares are not a primary source of coronal plasma. Significant progress in solving the coronal heating problem will require a coordination of approaches: observational studies, field-aligned hydrodynamic simulations, large-scale and localized 3D MHD simulations, and possibly also kinetic simulations. There is a unique value to each of these approaches, and the community must strive to coordinate better., Comment: Published by Phil. Trans. Royal Soc. A, 16 pages, 9 figures

Our knowledge of the diffuse emission that encompasses active regions is very limited. In the present paper we investigate two off-limb active regions, namely AR10939 and AR10961, to probe the underlying heating mechanisms. For this purpose we have used spectral observations from Hinode/EIS and employed the emission measure (EM) technique to obtain the thermal structure of these diffuse regions. Our results show that the characteristic EM distributions of the diffuse emission regions peak at log T = 6.25 and the cool-ward slopes are in the range 1.4 - 3.3. This suggests that both low as well as high frequency nanoflare heating events are at work. Our results provide additional constraints on the properties of these diffuse emission regions and their contribution to the background/foreground when active region cores are observed on-disk., Comment: 25 pages, 5 figures, Accepted for Publication in The Astrophysical Journal

Recent solar spectroscopic observations have shown that coronal spectral lines can exhibit asymmetric profiles, with enhanced emissions at their blue wings. These asymmetries correspond to rapidly upflowing plasmas at speeds exceeding ~ 50 km/s. Here, we perform a study of the density of the rapidly upflowing material and compare it to that of the line core which corresponds to the bulk of the plasma. For this task we use spectroscopic observations of several active regions taken by the Extreme Ultraviolet Imaging Spectrometer of the Hinode mission. The density sensitive ratio of the Fe XIV lines at 264.78 and 274.20 A is used to determine wing and core densities. We compute the ratio of the blue wing density to the core density and find that most values are of order unity. This is consistent with the predictions for coronal nanoflares if most of the observed coronal mass is supplied by chromospheric evaporation driven by the nanoflares. However, much larger blue wing-to-core density ratios are predicted if most of the coronal mass is supplied by heated material ejected with type II spicules. Our measurements do not rule out a spicule origin for the blue wing emission, but they argue against spicules being a primary source of the hot plasma in the corona. We note that only about 40% of the pixels where line blends could be safely ignored have blue wing asymmetries in both Fe XIV lines. Anticipated sub-arcsecond spatial resolution spectroscopic observations in future missions could shed more light on the origin of blue, red, and mixed asymmetries., Comment: ApJ, 2013, in press

It has previously been argued that 1. spicules do not provide enough pre-heated plasma to fill the corona, and 2. even if they did, additional heating would be required to keep the plasma hot as it expands upward. We here address the question of whether spicules play an important role by injecting plasma at cooler temperatures ($< 2$ MK), which then gets heated to coronal values at higher altitudes. We measure red-blue asymmetries in line profiles formed over a wide range of temperatures in the bright moss areas of two active regions. We derive emission measure distributions from the excess wing emission. We find that the asymmetries and emission measures are small and conclude that spicules do not inject an important (dominant) mass flux into the cores of active regions at temperatures $> 0.6$ MK ($\log T > 5.8$). These conclusions apply not only to spicules, but to any process that suddenly heats and accelerates chromospheric plasma (e.g., a chromospheric nanoflare). The traditional picture of coronal heating and chromospheric evaporation appears to remain the most likely explanation of the active region corona., Comment: 18 pages, 7 figures, Accepted for publication in ApJ

In this paper, the ability of the Hinode/EIS instrument to detect radiative signatures of coronal heating is investigated. Recent observational studies of AR cores suggest that both the low and high frequency heating mechanisms are consistent with observations. Distinguishing between these possibilities is important for identifying the physical mechanism(s) of the heating. The Differential Emission Measure (DEM) tool is one diagnostic that allows to make this distinction, through the amplitude of the DEM slope coolward of the coronal peak. It is therefore crucial to understand the uncertainties associated with these measurements. Using proper estimations of the uncertainties involved in the problem of DEM inversion, we derive confidence levels on the observed DEM slope. Results show that the uncertainty in the slope reconstruction strongly depends on the number of lines constraining the slope. Typical uncertainty is estimated to be about $\pm 1.0$, in the more favorable cases., Comment: 29 pages, 8 figures

A solar flare is composed of impulsive energy release events by magnetic reconnection, which forms and heats flare loops. Recent studies have revealed a two-phase evolution pattern of UV 1600\AA\ emission at the feet of these loops: a rapid pulse lasting for a few seconds to a few minutes, followed by a gradual decay on timescales of a few tens of minutes. Multiple band EUV observations by AIA further reveal very similar signatures. These two phases represent different but related signatures of an impulsive energy release in the corona. The rapid pulse is an immediate response of the lower atmosphere to an intense thermal conduction flux resulting from the sudden heating of the corona to high temperatures (we rule out energetic particles due to a lack of significant hard X-ray emission). The gradual phase is associated with the cooling of hot plasma that has been evaporated into the corona. The observed footpoint emission is again powered by thermal conduction (and enthalpy), but now during a period when approximate steady state conditions are established in the loop. UV and EUV light curves of individual pixels may therefore be separated into contributions from two distinct physical mechanisms to shed light on the nature of energy transport in a flare. We demonstrate this technique using coordinated, spatially resolved observations of UV and EUV emission from the footpoints of a C3.2 thermal flare.

One of the outstanding problems in all of space science is uncovering how the solar corona is heated to temperatures greater than 1 MK. Though studied for decades, one of the major difficulties in solving this problem has been unraveling the line-of-sight (LOS) effects in the observations. The corona is optically thin, so a single pixel measures counts from an indeterminate number (perhaps tens of thousands) of independently heated flux tubes, all along that pixel's LOS. In this paper we model the emission in individual pixels imaging the active region corona in the Extreme Ultraviolet. If LOS effects are not properly taken into account, erroneous conclusions regarding both coronal heating and coronal dynamics may be reached. We model the corona as a LOS integration of many thousands of completely independently heated flux tubes. We demonstrate that despite the superposition of randomly heated flux tubes, nanoflares leave distinct signatures in light curves observed with multi-wavelength and high time cadence data, such as those data taken with the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory. These signatures are readily detected with the time-lag analysis technique of Viall & Klimchuk (2012). Steady coronal heating leaves a different and equally distinct signature that is also revealed by the technique., Comment: 10 pages, 1 Table, and 4 Figures

The time-dependence of heating in solar active regions can be studied by analyzing the slope of the emission measure distribution cool-ward of the peak. In a previous study we showed that low-frequency heating can account for 0% to 77% of active region core emission measures. We now turn our attention to heating by a finite succession of impulsive events for which the timescale between events on a single magnetic strand is shorter than the cooling timescale. We refer to this scenario as a "nanoflare train" and explore a parameter space of heating and coronal loop properties with a hydrodynamic model. Our conclusions are: (1) nanoflare trains are consistent with 86% to 100% of observed active region cores when uncertainties in the atomic data are properly accounted for; (2) steeper slopes are found for larger values of the ratio of the train duration $\Delta_H$ to the post-train cooling and draining timescale $\Delta_C$, where $\Delta_H$ depends on the number of heating events, the event duration and the time interval between successive events ($\tau_C$); (3) $\tau_C$ may be diagnosed from the width of the hot component of the emission measure provided that the temperature bins are much smaller than 0.1 dex; (4) the slope of the emission measure alone is not sufficient to provide information about any timescale associated with heating - the length and density of the heated structure must be measured for $\Delta_H$ to be uniquely extracted from the ratio $\Delta_H/\Delta_C$.

Observational measurements of active region emission measures contain clues to the time-dependence of the underlying heating mechanism. A strongly non-linear scaling of the emission measure with temperature indicates a large amount of hot plasma relative to warm plasma. A weakly non-linear (or linear) scaling of the emission measure indicates a relatively large amount of warm plasma, suggesting that the hot active region plasma is allowed to cool and so the heating is impulsive with a long repeat time. This case is called {\it low-frequency} nanoflare heating and we investigate its feasibility as an active region heating scenario here. We explore a parameter space of heating and coronal loop properties with a hydrodynamic model. For each model run, we calculate the slope $\alpha$ of the emission measure distribution $EM(T) \propto T^\alpha$. Our conclusions are: (1) low-frequency nanoflare heating is consistent with about 36% of observed active region cores when uncertainties in the atomic data are not accounted for; (2) proper consideration of uncertainties yields a range in which as many as 77% of observed active regions are consistent with low-frequency nanoflare heating and as few as zero; (3) low-frequency nanoflare heating cannot explain observed slopes greater than 3; (4) the upper limit to the volumetric energy release is in the region of 50 erg cm$^{-3}$ to avoid unphysical magnetic field strengths; (5) the heating timescale may be short for loops of total length less than 40 Mm to be consistent with the observed range of slopes; (6) predicted slopes are consistently steeper for longer loops.